zbMATH — the first resource for mathematics

Duality property for positive weak Dunford-Pettis operators. (English) Zbl 1262.47057
The authors prove that an operator is weak Dunford-Pettis if its adjoint is, but the converse is in general false, and they produce some necessary and sufficient conditions under which each positive weak Dunford-Pettis operator has an adjoint which is weak Dunford-Pettis.
For basic terminologies and notations, the authors use [C. D. Aliprantis, Positive operators. Reprint of the 1985 original. Berlin: Springer (2006; Zbl 1098.47001)].

47B65 Positive linear operators and order-bounded operators
46B42 Banach lattices
47B07 Linear operators defined by compactness properties
Full Text: DOI
[1] C. D. Aliprantis and O. Burkinshaw, “Dunford-Pettis operators on Banach lattices,” Transactions of the American Mathematical Society, vol. 274, no. 1, pp. 227-238, 1982. · Zbl 0498.47013 · doi:10.2307/1999506
[2] N. J. Kalton and P. Saab, “Ideal properties of regular operators between Banach lattices,” Illinois Journal of Mathematics, vol. 29, no. 3, pp. 382-400, 1985. · Zbl 0568.47030
[3] B. Aqzzouz, K. Bouras, and A. Elbour, “Some generalizations on positive Dunford-Pettis operators,” Results in Mathematics, vol. 54, no. 3-4, pp. 207-218, 2009. · Zbl 1191.47051 · doi:10.1007/s00025-009-0372-2
[4] J. Diestel, “A survey of results related to the Dunford-Pettis property. Integration, topology and geometry in linear,” in Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces, vol. 2 of Contemporary Mathematics, pp. 15-60, Chapel Hill, NC, USA, 1980. · Zbl 0571.46013
[5] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, Dordrecht, The Netherlands, 2006, Reprint of the 1985 original. · Zbl 1098.47001
[6] B. Aqzzouz, A. Elbour, and A. W. Wickstead, “Positive almost Dunford-Pettis operators and their duality,” Positivity, vol. 15, no. 2, pp. 185-197, 2011. · Zbl 1232.46019 · doi:10.1007/s11117-010-0050-3
[7] B. Aqzzouz and K. Bouras, “Weak and almost Dunford-Pettis operators on Banach lattices,” preprint. · Zbl 1280.46010
[8] P. G. Dodds and D. H. Fremlin, “Compact operators in Banach lattices,” Israel Journal of Mathematics, vol. 34, no. 4, pp. 287-320, 1979. · Zbl 0438.47042 · doi:10.1007/BF02760610
[9] A. W. Wickstead, “Converses for the Dodds-Fremlin and Kalton-Saab theorems,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 120, no. 1, pp. 175-179, 1996. · Zbl 0872.47018 · doi:10.1017/S0305004100074752
[10] Z. L. Chen and A. W. Wickstead, “L-weakly and M-weakly compact operators,” Indagationes Mathematicae, vol. 10, no. 3, pp. 321-336, 1999. · Zbl 1028.47028 · doi:10.1016/S0019-3577(99)80025-1
[11] W. Wnuk, “Banach lattices with the weak Dunford-Pettis property,” Atti del Seminario Matematico e Fisico dell’Universit√† di Modena, vol. 42, no. 1, pp. 227-236, 1994. · Zbl 0805.46023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.